MinT - Best Known (89, 89+37, s)-Nets in Base 4

Best Known (89, 89+37, s)-Nets in Base 4

(89, 89+37, 384)-Net over F₄ — Constructive and digital

Digital (89, 126, 384)-net over F₄, using

trace code for nets [i] based on digital (5, 42, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(89, 89+37, 627)-Net over F₄ — Digital

Digital (89, 126, 627)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+37, 38178)-Net in Base 4 — Upper bound on s

There is no (89, 126, 38179)-net in base 4, because

1 times m-reduction [i] would yield (89, 125, 38179)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1809 886657 872668 812230 868526 053115 770278 417583 926039 288165 305060 069056 824118 > 4¹²⁵ [i]